Estimation of two ordered normal means under modified Pitman nearness criterion

Transcription

1 Ann Inst Stat Math (25) 67: DOI.7/s Estimation of two ordered normal means under modified Pitman nearness criterion Yuan-Tsung Chang Nobuo Shinozaki Received: 27 August 23 / Revised: 4 March 24 / Published online: 22 August 24 The Institute of Statistical Mathematics, Tokyo 24 Abstract The problem of estimating two ordered normal means is considered under the modified Pitman nearness criterion in the presence and absence of the order restriction on variances. When variances are not ordered, a class of estimators is considered that reduce to the estimators of a common mean when the unbiased estimators violate the order restriction. It is shown that the most critical case for uniform improvement with regard to the unbiased estimators is the one when two means are equal. When variances are ordered, a class of estimators is considered, taking the order restriction on variances into consideration. The proposed estimators of the mean with a larger variance improve upon the estimators that do not take the order restriction on variances into consideration. Although a similar improvement is not possible in estimating the mean with a smaller variance, a domination result is given in the simultaneous estimation. Keywords Order restriction Common mean Restricted MLE Unbiased estimator Pitman nearness Modified Pitman nearness Uniform improvement Introduction Estimation of restricted normal means (simple order, tree order or orthant restrictions etc.) has been considered by many authors and some estimators improving upon the Y.-T. Chang (B) Department of Social Information, Faculty of Studies on Contemporary Society, Mejiro University, 4-3- Nakaochiai, Shinjuku-ku, Tokyo , Japan chogenso@gmail.com; cho@mejiro.ac.jp N. Shinozaki Faculty of Science and Technology, Keio University, 3-4- Hiyoshi, Kohoku-ku, Yokohama , Japan

2 864 Y.-T. Chang, N. Shinozaki unbiased estimators have been proposed. Most works related to the inference about the restricted parameters are reviewed by Barlow et al. (972), Robertson et al. (988), Silvapulle and Sen (24) and Van Eeden (26). As a criterion to evaluate the goodness of estimators, mean squared error (MSE) is usually used and in some cases stochastic domination as well. A possible alternative criterion, which was introduced by Pitman (937), is Pitman nearness. Let T and T 2 be two estimators of θ. Then, Pitman nearness of T relative to T 2 is defined by PN θ (T, T 2 ) = P{ T θ < T 2 θ } and T is said to be closer to θ than T 2 if PN θ (T, T 2 ) /2 for any parameter value. We refer the reader to Rao (98), Keating and Mason (985, 986), Peddada (985), Rao et al. (986), Peddada and Khattree (986) for its discussions. Many works related to Pitman nearness were published in the special issue of Communications in Statistics Theory and Methods A2 () in 992 and were unified in the monograph by Keating et al. (993). Although the Pitman nearness has been severely criticized as a measure of comparing estimators (see Robertson et al. 993), we believe that it is a useful criterion in comparing two estimators and understanding the nature of the estimators. Here, we consider the estimation of two ordered normal means when order restriction on the unknown variances is present and not present. We propose some estimators of the means which improve upon some baseline estimators which do not take into account of the order restrictions in terms of modified Pitman nearness criterion suggested by Gupta and Singh (992). We first state some fundamental results on the estimation of a common mean and ordered means when MSE, stochastic domination or Pitman nearness is concerned. Let X ij, i =, 2, j =,...,n i be independent observations from normal distribution with mean μ i and variance σi 2, where both μ i and σi 2 are unknown. Let n i X i = X ij /n i and si 2 = (X ij X i ) 2 /(n i ) j= be the unbiased estimators of μ i and σ 2 i, respectively. For the common mean problem (μ = μ 2 = μ) when two variances are unknown and there is no order restriction on the two variances, Graybill and Deal (959) proposed the estimator n i j= ˆμ GD n s2 2 = n s2 2 + n 2s 2 n 2 s X 2 + n s2 2 + n 2s 2 X 2 and gave a necessary and sufficient condition on n and n 2 for ˆμ GD to have a smaller variance than both X and X 2. Later, many authors, including Brown and Cohen (974), Khatri and Shah (974), and Bhattacharya (98), have given a class of improved estimators of the form

3 Estimation of two ordered normal means under modified Pitman nearness 865 ˆμ(γ ) = γ X + ( γ) X 2, where γ is a function of s 2, s2 2 and possibly ( X X 2 ) 2. As for Pitman nearness criterion, Kubokawa (989) has given a broad class of estimators of a common mean with γ given by a γ ψ = + Rψ(s 2, s2 2,( X X 2 ) 2 ) where R = (bs2 2 + c( X X 2 ) 2 )/s 2, ψ is a positive valued function and a, b, and c are nonnegative constants. For suitably chosen ψ,a, b, and c, Kubokawa (989) has given a sufficient condition on n and n 2 so that ˆμ(γ ψ ) is closer to μ than X. Thus, Kubokawa (989) has shown that ˆμ GD is closer to μ than both X and X 2 if n 5 and n 2 5. Misra and Van der Meulen (997) have discussed estimating a common mean of k( 2) normal distributions when order restriction is given on variances and proposed an estimator improving upon ˆμ GD in terms of stochastic dominance and Pitman nearness. For the case when k normal means satisfy simple order restriction and variances are known, Lee (98) has shown that restricted MLE (maximum likelihood estimator) uniformly improves upon sample means in terms of MSE. Kelly (989) and Hwang and Peddada (994) have proven that restricted MLE universally dominates sample means. For the case when σi 2 s are unknown and two means satisfy the order restriction, μ μ 2, Oono and Shinozaki (25) have proposed the estimators of μ i, i =, 2 { ˆμ OS = min X, ˆμ GD} { and ˆμ OS 2 = max X 2, ˆμ GD}, and have shown that ˆμ i OS uniformly improves upon X i in terms of MSE if and only if MSE of ˆμ i OS is not larger than that of X i for the case when μ = μ 2. Further, it has been shown that ˆμ i OS improves upon X i if and only if ˆμ GD improves upon X i in the common mean problem. It should be mentioned that Garren (2) has proposed a similar estimator with si 2 replaced by the MLE of σi 2 in ˆμ i GD and has given a condition on n and n 2 for the estimator to have larger (or smaller) MSE when μ = μ 2 and σ2 2/σ 2 is sufficiently large. Next, we consider the estimation of two ordered means when order restriction, σ 2 σ 2 2, is given on two variances. Such a situation can occur, for example, if there are two kinds of motor engines, one of which is developed by a new method, and has higher power but has larger variation than the other one developed by a standard method. Shi (994) and Ma and Shi (22) have discussed the order restricted MLE of μ i and σi 2. Chang et al. (22) have considered a class of estimators of μ i, i =, 2 of the form ˆμ (γ ) = min{ X,γ X +( γ) X 2 }, ˆμ 2 (γ ) = max{ X 2,γ X +( γ) X 2 }, () where γ is a function of s 2, s2 2, and ( X X 2 ) 2. γ may be considered to be an estimator of n σ 2 2 /(n σ n 2σ 2 ) which is not larger than n /(n + n 2 ) when σ 2 σ 2 2. Thus,

4 866 Y.-T. Chang, N. Shinozaki as in the common mean problem (see, for example, Seshadri 966; Nair 982; Elfessi and Pal 992), we may improve upon ˆμ i (γ ) by replacing γ with γ + = { γ, if γ n n +n 2, γ, if γ< n n +n 2, (2) and choosing γ appropriately. Actually, Chang et al. (22) have shown that ˆμ 2 (γ + ) stochastically dominates ˆμ 2 (γ ) if γ is chosen to satisfy n n + n 2 γ 2 n n + n 2 γ, when γ< n n + n 2. (3) Further, Chang et al. (22) have shown that although ˆμ (γ ) has smaller MSE than ˆμ (γ + ) for sufficiently large μ 2 μ,(ˆμ (γ + ), ˆμ 2 (γ + )) dominates ( ˆμ (γ ), ˆμ 2 (γ )) in the sense that { 2 ( ˆμi (γ + ) 2 { 2 ( ) ) μ i ˆμi (γ ) μ 2 i P d} P d} i= τ i for any d >, where τi 2 = σi 2/n i. Now, we state a modification of Pitman nearness to compare the estimators when they are equal with positive probability. Nayak (99) defined modified Pitman nearness of an estimator T of θ relative to the other estimator T 2 by i= MPN θ (T, T 2 ) = P{ T θ < T 2 θ T = T 2 }. If MPN θ (T, T 2 ) /2 for any parameter value, then T is said to be closer to θ than T 2. Gupta and Singh (992) have applied modified Pitman nearness to the estimation of ordered means of two normal population with common variance and have shown that MLE is closer than the unbiased estimator. Here, we consider the estimation problems under modified Pitman nearness for the following two cases Case-I: estimation of two ordered means when σ 2 and σ 2 2 are unrestricted, Case-II: estimation of two ordered means when σ 2 σ 2 2. We are much interested in whether or not the similar results can be obtained under the modified Pitman nearness criterion to those obtained by Oono and Shinozaki (25) and Chang et al. (22) under MSE or stochastic domination criterion. We first treat Case-I in Sect. 2 of this paper. With respect to modified Pitman nearness, we show that ˆμ i (γ ) improves upon X i if and only if MPN μi ( ˆμ i (γ ), X i ) /2 when μ = μ 2, which is the most critical case for uniform improvement. Further, it is shown that ˆμ i (γ ) improves upon X i if and only if ˆμ(γ ) improves upon X i for the same γ in estimating a common mean. Thus, the problem of improving upon the unbiased estimators of two ordered means essentially reduces to the one of a common mean. This conclusion also applies even when variances are ordered. However, we τ i

5 Estimation of two ordered normal means under modified Pitman nearness 867 show that when σ 2 σ 2 2, ˆμGD improves upon X 2 for a wide range of (n, n 2 ) in Sect. 3. In Sect. 4, for Case-II, we treat the problem of improving upon ˆμ i (γ ) by replacing γ with γ + satisfying (2) and (3) with respect to modified Pitman nearness when the restrictions μ μ 2 and σ 2 σ 2 2 are given. We show that ˆμ 2(γ + ) improves upon ˆμ 2 (γ ) for suitably chosen γ,but ˆμ (γ + ) does not improve upon ˆμ (γ ) even if γ is chosen appropriately. Simultaneous estimation of μ and μ 2 is also discussed with respect to generalized Pitman nearness based on sum of normalized squared errors (Rao et al. 986; Peddada and Khattree 986). In Sect. 5, we give some results of the numerical study to evaluate the performance of the proposed estimators for the choice γ = n s2 2/(n s2 2 + n 2s 2 ) in terms of modified Pitman nearness. In Sect. 6, wemake a conclusion. Throughout this paper, we assume that γ is a function of s 2 and s2 2 satisfying γ and that in some places γ may depend on ( X X 2 ) 2 as well. 2 Case-I: improving unbiased estimators of ordered means when variances are unrestricted In this section, we treat the estimation problem of ordered means μ μ 2 when variances are not ordered. We consider estimators of μ i of the form () and compare them with the unbiased estimator X i. We first show that for the case when γ is a function of s 2 and s2 2 only, the most critical case for ˆμ i(γ ) to be closer to μ i than X i is the one when μ = μ 2. Further, it is shown that ˆμ i (γ ) improves upon X i if and only if ˆμ(γ ) dominates X i in the estimation problem of a common mean. Theorem Suppose that γ is a function of s 2 and s2 2. (i) MPN μi ( ˆμ i (γ ), X i ) /2 for all μ μ 2 and for all σ 2 and σ 2 2 if and only if for all σ 2 and σ 2 2, MPN μ i ( ˆμ i (γ ), X i ) /2 when μ = μ 2. (ii) MPN μi ( ˆμ i (γ ), X i ) /2 for all μ μ 2 and for all σ 2 and σ 2 2 if and only if for all σ 2 and σ 2 2, PN μ( ˆμ(γ ), X i ) /2 to estimate μ when μ = μ 2 = μ. Remark In the estimation problem of a common mean, Kubokawa (989) has given a sufficient condition on sample sizes n and n 2 for ˆμ(γ ) to be closer to μ than X i for some specified class of γ. For example, for the choice of γ = n s 2 2 /(n s n 2s 2 ), ˆμ GD is closer to μ than both X and X 2 if n 5 and n 2 5. Proof We need only to give a proof for the case of μ. (i) Since ˆμ (γ ) = X if and only if X > X 2 and γ<, we have MPN μ ( ˆμ (γ ), X ) = P{ ˆμ (γ ) μ < X μ ˆμ (γ ) = X } = P{ γ X + ( γ) X 2 μ < X μ X > X 2,γ <} = P{2μ <( + γ) X + ( γ) X 2 X > X 2,γ <} = P{ <( + γ)z + ( γ)z 2 Z > Z 2,γ <}, (4)

6 868 Y.-T. Chang, N. Shinozaki where Z i = X i μ, i =, 2 are distributed as N(,τ 2) and N(Δ, τ 2 2 ), respectively, Δ = μ 2 μ and τi 2 = σi 2/n i. Now, we fix the values of s 2 and s2 2 for which γ<and consider the conditional probability P{ <( + γ)z + ( γ)z 2 Z > Z 2, s 2, s2 2 } f (Δ) as a function of Δ. We need only to show that f (Δ) f (). Putting d = ( + γ)/( γ), we define the sets A ={(z, z 2 ) z z 2, z 2 dz }, B ={(z, z 2 ) z z 2, z 2 < dz }, A ={(z, z 2 ) z z 2, z 2 } and A 2 ={(z, z 2 ) z 2 dz, z 2 < }. Since A and A 2 are disjoint and A = A A 2,wehave P Δ (A) f (Δ) f () = P Δ (A)+ P Δ (B) P (A) P (A)+ P (B) = {P Δ(A )P (B) P (A )P Δ (B)}+{P Δ (A 2 )P (B) P (A 2 )P Δ (B)}, {P Δ (A)+ P Δ (B)}{P (A)+ P (B)} where, for example, P (A) denotes the probability of the set A when Δ =. We first show that P Δ (A )P (B) P (A )P Δ (B) >forδ>. For that purpose, we note that { } P Δ (B) = exp (z 2 Δ) 2 z2 /d φ(z /τ )dz dz 2 2πτ2 τ < exp { Δ2 2τ2 2 = exp { Δ2 2τ2 2 Similarly, we have } 2τ 2 2 2πτ2 exp { z 2 z2 2 2τ 2 2 } z2 /d z 2 τ φ(z /τ )dz dz 2 } P (B). (5) { } P Δ (A ) = exp (z 2 Δ) 2 2πτ2 2τ2 2 φ(z /τ )dz dz 2 z 2 τ { } > exp Δ2 2τ2 2 P (A ). (6) From (5) and (6), we see that P Δ (A )P (B) P (A )P Δ (B) >. Next, we show that P Δ (A 2 )P (B) P (A 2 )P Δ (B) >forδ>. We express P Δ (A 2 ) as

7 Estimation of two ordered normal means under modified Pitman nearness 869 { } P Δ (A 2 ) = exp (z 2 Δ) 2 2πτ2 2τ2 2 φ(z /τ )dz dz 2 z 2 /d τ = P Δ {Z 2 < }E Δ [g(z 2 ) Z 2 < ], where g(z 2 ) = z 2 /d φ(z /τ )/τ dz. Since g(z 2 ) is an increasing function and the conditional distribution of Z 2 < is stochastically smallest when Δ =, we have for Δ> P Δ (A 2 )>P Δ {Z 2 < }E [g(z 2 ) Z 2 < ] =P (A 2 )P Δ {Z 2 < }/P {Z 2 < }. (7) Similarly, since h(z 2 ) = z 2 /d z 2 φ(z /τ )/τ dz is a decreasing function, we have { } P Δ (B) = exp (z 2 Δ) 2 z2 /d 2πτ2 2τ2 2 φ(z /τ )dz dz 2 z 2 τ = P Δ {Z 2 < }E Δ [h(z 2 ) Z 2 < ] < P Δ {Z 2 < }E [h(z 2 ) Z 2 < ] = P (B)P Δ {Z 2 < }/P {Z 2 < }. (8) From (7) and (8), we have P Δ (A 2 )P (B) P (A 2 )P Δ (B) > and we have shown that f (Δ) > f () for Δ>. (ii) In the estimation problem of a common mean, as is stated in Kubokawa (989,p. 48), Eq. (8), ˆμ(γ ) is closer to μ than X if and only if P{( γ)(u 2 U ) 2 + 2U (U 2 U ) } /2, (9) where U i = X i μ, i =, 2. Since ( γ)(u 2 U ) 2 + 2U (U 2 U ) = (U 2 U ){( + γ)u + ( γ)u 2 }, the LHS of (9) is expressed as We notice that P{U U 2 }P{( + γ)u + ( γ)u 2 < U U 2 } + P{U > U 2 }P{( + γ)u + ( γ)u 2 > U > U 2 }. P{( + γ)u + ( γ)u 2 < U U 2 } = P{( + γ)u + ( γ)u 2 > U > U 2 }, since U and U 2 are symmetrically distributed about the origin. Thus, noting that P{U U 2 }=P{U > U 2 }=/2, we see that the LHS of (9) is equal to P{( + γ)u + ( γ)u 2 > U > U 2 },

8 87 Y.-T. Chang, N. Shinozaki which is MPN μ ( ˆμ (γ ), X ) given by (4) for the case μ = μ 2. Therefore, we see from (i) that MPN μ ( ˆμ (γ ), X )) /2 for all μ μ 2 and for all σ 2 and σ 2 2 if and only if PN μ ( ˆμ(γ ), X i ) /2 for all μ and for all σ 2 and σ 2 2. We complete the proof. Remark 2 We should mention about the general case when γ is a function of s 2, s2 2 and ( X X 2 ) 2. We first consider the case when we estimate μ and suppose that ˆμ (γ ) is closer to μ than X, where γ is a function of s 2, s2 2 and possibly ( X X 2 ) 2. For any γ satisfying γ γ < ifγ <, ˆμ (γ ) is closer to μ than X.This is seen from (4), since (4) istrueevenwhenγ depends on ( X X 2 ) 2 and (4) is an increasing function of γ. Next, we consider the case when we estimate μ 2 and suppose that ˆμ 2 (γ ) is closer to μ 2 than X 2, where γ is a function of s 2, s2 2 and possibly ( X X 2 ) 2. For any γ satisfying <γ γ if γ >, ˆμ 2 (γ ) is closer to μ 2 than X 2. 3 Case-II: improving unbiased estimators of ordered means when variances are ordered Now, we treat the estimation problem of ordered means μ μ 2 when the order restriction σ 2 σ 2 2 on variances is present. It is necessary for us to modify the estimators ˆμ i (γ ), i =, 2 treated in Sect. 2, taking into account of the order restriction σ 2 σ 2 2. We discuss such modifications of estimators in the next section. Before that we discuss a sufficient condition on the sample sizes n and n 2 for ˆμ i (γ ) to improve upon X i when the order restriction σ 2 σ 2 2 is present. It is easily seen that Theorem is true even for the case when the restriction σ 2 σ 2 2 is present, if we replace for all σ 2 and σ 2 2 by forallσ 2 σ 2 2. That is, the most critical case for ˆμ i (γ ) to improve upon X i is the case when μ = μ 2, and ˆμ i (γ ) improves upon X i if and only if ˆμ(γ ) improves upon X i for the same γ in the estimation of a common mean. Thus, the problem reduces to the estimation problem of a common mean when the restriction σ 2 σ 2 2 is present, which we are concerned with in the rest of this section. For the case when the restriction σ 2 σ 2 2 is not present, Kubokawa (989) has given a sufficient condition on n and n 2 which guarantees that ˆμ(γ ) improves upon X i for some specified class of γ. We modify Kubokawa (989) s discussion for the case when the restriction σ 2 σ 2 2 is present. We first note that there is no room for modifying the sufficient condition for ˆμ(γ ) to improve upon X which is associated with smaller variance σ 2, since the case σ 2/σ 2 2 is critical as is seen from the proof of Kubokawa (989). However, we show that ˆμ(γ ) improves upon X 2 which is associated with larger variance for a wide range of (n, n 2 ). Unfortunately, we have not succeeded in showing generally the monotonicity of PN μ ( ˆμ(γ ), X 2 ) in σ 2/σ 2 2. Here, we restrict ourselves to the estimator ˆμ(γ ) = γ X + ( γ) X 2 with γ of the form n s2 2 γ = n s2 2 + cn 2s 2, ()

9 Estimation of two ordered normal means under modified Pitman nearness 87 where c is a positive constant to be chosen properly. From Remark 2, it is seen that if γ = n s2 2/(n s2 2 + cn 2s 2 ) gives an improved estimator, then any γ satisfying <γ n s2 2/(n s2 2 + cn 2s 2 ) also gives an improvement even when γ depends on ( X X 2 ) 2. We essentially follow the notations given in the proof of Kubokawa (989) and put τ = (n σ2 2)/(n 2σ 2 ) and Z = τ + cf, where F is a random variable having F-distribution with degrees of freedom (n, n 2 ). Putting G = τ{ ( + τ)/(2z)},pn μ ( ˆμ(γ ), X 2 ) /2 if h(τ) = E [ (G+ G 2 +) 2 ] f, (v)dv /2, where f, ( ) is a density of an F-distribution with degrees of freedom (, ) and the expectation is taken with respect to the random variable F. Further, the derivative of h(τ) is given in Kubokawa (989) as h (τ) = C 2 τ( + τ) E [ 2Z 2 ( + 3τ)Z + 2τ( + τ) 4Z 2 4τ Z + τ( + τ) ], () where C is a positive constant. First, we see that for any c >, h(τ) /2 forτ, since τ implies G.τ for any σ 2 2 σ 2 if and only if n n 2. Thus, we see that ˆμ(γ ) with γ given by () improves upon X 2 if n n 2. Kubokawa (989) has shown that if c (n )/{2(n 3)} for n 4, h (τ) for any τ and thus h(τ) h() = /2, for any τ. Therefore, we see that under the same condition on c, h(τ) /2 for any τ n /n 2. Since (n )/{2(n 3)} forn 5, ˆμ GD improves upon X 2 for n 5. Even when n 4, we can show that ˆμ(γ ) improves upon X 2 for a wide range of n 2.From(), we see that h (τ) if2z 2 (+3τ)z +2τ(+τ) for any z τ, which is true if τ ( )/7 =.938. Therefore, h (τ) forτ n /n 2 if n /n Thus, we have seen that if n /n and h(n /n 2 ) /2, ˆμ(γ ) improves upon X 2. Setting c =, we have numerically evaluated h(n /n 2 ) for some values of (n, n 2 ) as given in Table. From Table, we see that ˆμ GD is closer to μ than X 2 for (n = 2, n 2 7), (n = 3, n 2 26) and (n = 4, n 2 3), and is not closer to μ than X 2 for (n = 2, 8 n 2 3) and (n = 3, 27 n 2 3). 4 Case-II: a class of improved estimators of ordered means when variances are also ordered In Sect. 2, we have compared ˆμ i (γ ) with X i for Case-I, where the order restriction is given on two means and σ 2 and σ 2 2 are unrestricted. Although we also compared them for Case-II, where order restrictions are given on means and variances in Sect. 3, ˆμ i (γ ) is not constructed so that the order restriction on variances is taken into consideration. Since the inequality σ 2 σ 2 2 implies n σ2 2/(n σ2 2 + n 2σ 2) n /(n + n 2 ),wemay modify ˆμ i (γ ) so that γ n /(n +n 2 ) is satisfied. As is given in (3), we replace γ by

10 872 Y.-T. Chang, N. Shinozaki Table h(n /n 2 ) n 2 /n γ n /(n + n 2 ) when γ<n /(n + n 2 ).Usingγ + given in (2), we have ˆμ i (γ + ) and compare it with ˆμ i (γ ) in this Section. In Theorem 2, for the estimation of the mean μ 2 of the population with larger variance, we show that ˆμ 2 (γ + ) is closer to μ 2 than ˆμ 2 (γ ) for suitably chosen γ. In Theorem 3, for the estimation of the mean μ of the population with smaller variance, we show that MPN μ ( ˆμ (γ + ), ˆμ (γ )) < /2 for some parameter values even when γ is chosen appropriately. In Theorem 4, we show that improvement is possible when we estimate means (μ,μ 2 ) simultaneously under generalized Pitman nearness based on sum of normalized squared errors. Here, we assume that γ may depend on ( X X 2 ) 2 as well.

11 Estimation of two ordered normal means under modified Pitman nearness Estimation of individual means when means and variances are ordered Theorem 2 Suppose that P{ X X 2 >,γ <n /(n + n 2 )} > for any μ μ 2 and σ 2 σ 2 2.If γ is chosen so that n n + n 2 γ 2 n n + n 2 γ when γ< n n + n 2, (2) then for any μ μ 2 and σ 2 σ 2 2, MPN μ 2 ( ˆμ 2 (γ + ), ˆμ 2 (γ )) /2. Proof We show that P { ˆμ 2 (γ + ) μ 2 < ˆμ 2 (γ ) μ 2, ˆμ 2 (γ + ) = ˆμ 2 (γ ) } P { ˆμ 2 (γ + ) = ˆμ 2 (γ ) } /2. (3) From (2), we see that ˆμ 2 (γ + ) = ˆμ 2 (γ ) if and only if X > X 2 and γ<n /(n +n 2 ). In this case, ˆμ 2 (γ + ) = γ X + ( γ) X 2 >γ X + ( γ) X 2 =ˆμ 2 (γ ). Thus, the LHS of (3) is expressed as P { ˆμ 2 (γ + ) μ 2 < ˆμ 2 (γ ) μ 2, X > X 2,γ <n /(n + n 2 ) } = P { ˆμ 2 ( γ)+ˆμ 2 (γ ) < 2μ 2, X > X 2,γ <n /(n + n 2 ) } = P { (γ + γ)( X μ 2 ) +{2 (γ + γ)}( X 2 μ 2 )<, X > X 2,γ<n /(n + n 2 ) }. (4) Now, we make the variable transformation V = ( X μ 2 ) ( X 2 μ 2 ), V 2 = (τ 2 2 /τ 2 )( X μ 2 ) + ( X 2 μ 2 ). Then, V and V 2 are independently distributed as N ( Δ, τ 2 + τ 2 2 ) and N( (τ 2 2 /τ 2 )Δ, τ 2 2 (τ 2 + τ 2 2 )/τ 2 ), respectively, where Δ = μ 2 μ. Noting that X μ 2 = τ 2 (V + V 2 ) τ 2 + τ 2 2 and X 2 μ 2 = τ 2V 2 τ2 2V τ 2 + τ 2 2, and denoting the indicator functions of the sets V > and γ < n /(n + n 2 ) by I V > and I γ<n /(n +n 2, respectively, (4) becomes {[ P (γ + γ)τ 2 {2 (γ + γ)}τ 2 2 = E Φ [{2 (γ + γ)}τ 2 2 (γ + γ)τ2 ]V +2τ 2 2 Δ ] 2τ τ 2 τ 2 +τ 2 2 } V +2τ 2 V 2 <, V >,γ <n /(n +n 2 ) I V > I γ<n /(n +n 2 ), (5)

12 874 Y.-T. Chang, N. Shinozaki where expectation is taken with respect to V, s 2 and s2 2. Noting that {2 (γ + γ)}τ 2 2 (γ + γ)τ 2 since γ + γ 2n /(n + n 2 ),wehave (5) P {V >,γ <n /(n + n 2 )} /2, which is RHS of (3). Since the inequality (3) implies that MPN μ2 ( ˆμ 2 (γ + ), ˆμ 2 (γ )) /2, this completes the proof. Remark 3 We may mention that if we choose γ = 2n /(n + n 2 ) γ, γ n /(n + n 2 ) = γ n /(n + n 2 ). We note that 2n /(n + n 2 ) γ may be larger than for some values of s 2 and s2 2 if n > n 2. In such cases, we should replace γ by γ = min( γ,), which leads to a further improvement. Remark 4 For the case when γ, we apply Theorem 2 and we obtain a class of estimators improving upon X 2. We see that for any γ satisfying n /(n + n 2 ) γ 2n /(n + n 2 ), ˆμ 2 (γ ) improves upon X 2. From Remark 2, we can broaden the class of improved estimators. We see that when n n 2, ˆμ 2 (γ ) improves upon X 2 for any γ satisfying γ. Remark 5 Suppose that ˆμ 2 (γ ) improves upon X 2 under modified Pitman nearness criterion. Although ˆμ 2 (γ + ) improves upon ˆμ 2 (γ ) for suitably chosen γ from Theorem 2, this does not imply that ˆμ 2 (γ + ) improves upon X 2 since transitivity does not hold for Pitman nearness criterion. As a matter of fact, MPN μ2 ( ˆμ 2 (γ ), X 2 )> MPN μ2 ( ˆμ 2 (γ + ), X 2 ) since ˆμ 2 (γ + ) μ 2 < X 2 μ 2 implies ˆμ 2 (γ ) μ 2 < X 2 μ 2 for the case when X > X 2 and γ<n /(n + n 2 ). Thus, ˆμ 2 (γ + ) is less likely to improve upon X 2 than ˆμ 2 (γ ). We need to examine whether ˆμ 2 (γ + ) improves upon X 2 or not in estimating a common mean, from the results given in Sects. 2 and 3 for the case when n 2 > n. Remark 6 When we consider the estimation problem of a common mean of two normal distributions when variances satisfy the order restriction σ 2 σ 2 2 and consider the estimator of the form ˆμ(γ ) = γ X + ( γ) X 2, then PN μ ( ˆμ(γ + ), ˆμ(γ )) /2 if γ satisfies the condition (2). We can show this by treating the cases X X 2 and X > X 2 separately along the same line as the proof of Theorem 2. In the next Theorem, we treat the estimation problem of the mean μ which is associated with smaller variance and show that MPN μ ( ˆμ (γ + ), ˆμ (γ )) < /2 when the difference of the two means, Δ = μ 2 μ, is sufficiently large. To prove it, we need the following. Lemma Let g( ) be the probability density function of the normal distribution N( Δ, σ 2 ). For any c > and d > for sufficiently large Δ. A proof is given in Appendix A. Φ(cy dδ)g(y)dy < g(y)dy/2

13 Estimation of two ordered normal means under modified Pitman nearness 875 Theorem 3 Suppose that γ is a non-decreasing function of X X 2 > and that P{ X X 2 >,γ < n /(n + n 2 )} > for any μ μ 2 and σ 2 σ 2 2.If γ is chosen so that (2) is satisfied, then for sufficiently large Δ, MPN μ ( ˆμ (γ + ), ˆμ (γ )) < /2. Proof Noticing again that ˆμ (γ + ) = ˆμ (γ ) if and only if X > X 2 and γ<n /(n + n 2 ) and that in this case ˆμ (γ + )> ˆμ (γ ),wehave P{ ˆμ (γ + ) μ < ˆμ (γ ) μ, ˆμ (γ + ) = ˆμ (γ )} = P{ˆμ (γ ) +ˆμ ( γ)<2μ, X > X 2,γ <n /(n + n 2 )} = P { (γ + γ)( X μ ) +{2 (γ + γ)}( X 2 μ )<, X > X 2,γ <n /(n + n 2 ) }. (6) We make the variable transformation Y = ( X μ ) ( X 2 μ ) and Y 2 = ( X μ ) + (τ 2 /τ 2 2 )( X 2 μ ). Then, Y and Y 2 are independently distributed as N( Δ, τ 2 + τ 2 2) and N((τ 2/τ 2 2)Δ, τ 2(τ 2 + τ 2 2)/τ 2 2 ), respectively. Noting that X μ = τ 2 Y + τ 2 2 Y 2 τ 2 + τ 2 2 and X 2 μ = τ 2 2(Y 2 Y ) τ 2 + τ 2 2, (6) is rewritten as P { [(γ + γ)τ 2 {2 (γ + γ)}τ 2 2 ]Y +2τ2 2 Y 2 <, Y >,γ <n /(n +n 2 ) } = E Φ [{2 (γ + γ)}τ 2 2 (γ + γ)τ2 ]y /2 τ 2Δ I γ<n τ τ 2 τ 2+τ 2 2 /(n +n 2 )g(y )dy, (7) where expectation is taken with respect to s 2 and s2 2 and g( ) is the density function of N( Δ, τ 2 + τ 2 2 ). Setting we have c = τ 2 /(τ τ 2 + τ 2 2 ) and d = τ /(τ 2 τ 2 + τ 2 2 ), Φ [{2 (γ + γ)}τ 2 2 (γ + γ)τ2 ]y /2 τ 2Δ Φ(cy dδ). τ τ 2 τ 2 + τ 2 2

14 876 Y.-T. Chang, N. Shinozaki Fixing s 2 and s2 2, we define t(s 2, s2 2 ) = sup{y = X X 2 > γ<n /(n + n 2 )}. Since γ is a non-decreasing function of X X 2, Φ(cy dδ)i γ<n /(n +n 2 )g(y )dy = Thus, we need only to show that t(s 2,s 2 2 ) Φ(cy dδ)g(y )dy < /2 t(s 2,s 2 2 ) From Lemma, we see that for sufficiently large Δ, which implies t Φ(cy dδ)g(y )dy < Φ(cy dδ)g(y )dy < t t(s 2,s 2 2 ) Φ(cy dδ)g(y )dy. g(y )dy /2, g(y )dy /2 g(y )dy. for any t > since Φ(cy dδ) is an increasing function of y >. This completes the proof. Remark 7 Although ˆμ (γ + ) does not dominate ˆμ (γ ) from Theorem 3, we see that if ˆμ (γ ) dominates X and P{ X > X 2,γ < n /(n + n 2 )} >, then ˆμ (γ + ) dominates X for any γ satisfying the condition (2). This is seen from (4) since the probability of (4) is an increasing function of γ. Remark 8 From the expression (7) of the probability P{ ˆμ (γ + ) μ < ˆμ (γ ) μ, ˆμ (γ + ) = ˆμ (γ )}, we see that the probability is larger than P{ˆμ (γ + ) = ˆμ (γ )}/2 when Δ = and σ 2 /σ 2 2 is sufficiently small. Thus, neither ˆμ (γ + ) nor ˆμ (γ ) dominates the other. 4.2 Simultaneous estimation of ordered means when variances are also ordered Here, we simultaneously estimate ordered means μ μ 2 when variances are also ordered. We show that ˆµ(γ + ) = ( ˆμ (γ + ), ˆμ 2 (γ + )) improves upon ˆµ(γ ) = ( ˆμ (γ ), ˆμ 2 (γ )) under generalized Pitman nearness based on sum of normalized squared errors in the following. Theorem 4 Suppose that P{ X X 2 >,γ <n /(n + n 2 )} > for any μ μ 2 and σ 2 σ 2 2.If γ is chosen so that (2) is satisfied, then for any μ μ 2 and σ 2 σ 2 2, MPN µ( ˆµ(γ + ), ˆµ(γ )) > /2, where

15 Estimation of two ordered normal means under modified Pitman nearness 877 MPN µ ( ˆµ(γ + ), ˆµ(γ )) { 2 = P {ˆμ i (γ + ) μ i } 2 /τi 2 i= } 2 {ˆμ i (γ ) μ i } 2 /τi 2 ˆµ(γ + ) = ˆµ(γ ). i= Proof Putting Z i = X i μ, i =, 2 and Δ = μ 2 μ, we express sum of normalized squared errors for ˆµ(γ ) as 2 i= {ˆμ i (γ ) μ i } 2 /τ 2 i = τ 2 = τ 2 ( = τ 2 { γ X + ( γ) X 2 μ } 2 + {γ Z + ( γ)z 2 } 2 + ) + τ2 2 τ 2 2 τ 2 2 {γ Z + ( γ)z 2 } 2 2Δ τ 2 2 { γ X + ( γ) X 2 μ 2 } 2 {γ Z + ( γ)z 2 Δ} 2 {γ Z + ( γ)z 2 }+ Δ2 τ2 2. (8) We note that ˆµ(γ + ) = ˆµ(γ ) if and only if X > X 2 and γ < n /(n + n 2 ).In this case, by replacing γ by γ in (8), we have sum of normalized squared errors for ˆµ(γ + ) as ( 2 {ˆμ i (γ + ) μ i } 2 /τi 2 = i= Thus, we have 2 {ˆμ i (γ ) μ i } 2 /τi 2 i= 2 i= [( = (γ γ)(z Z 2 ) τ 2 2Δ τ 2 2 {ˆμ i (γ + ) μ i } 2 /τ 2 i τ 2 which is non-negative if and only if ) + τ2 2 { γ Z + ( γ)z 2 } 2 { γ Z + ( γ)z 2 }+ Δ2 τ2 2. ) ] + τ2 2 [(γ + γ)z +{(2 (γ + γ)}z 2 ] 2Δ τ2 2, (τ 2 + τ 2 2 ) [ (γ + γ)z +{2 (γ + γ)}z 2 ] 2τ 2 Δ since Z Z 2 = X X 2 > and γ >γ. Therefore, we need only to show that { P (τ 2 + τ 2 2 )[(γ + γ)z +{2 (γ + γ)}z 2 ] 2τ 2 Δ,

16 878 Y.-T. Chang, N. Shinozaki } Z > Z 2,γ <n /(n + n 2 ) P {Z > Z 2,γ <n /(n + n 2 )} /2. Now, we make the variable transformation Y = Z Z 2, Y 2 = Z + (τ 2 /τ 2 2 )Z 2. Then, Y and Y 2 are the same with those given in the proof of Theorem 3. Since Y and Y 2 are independently distributed as N( Δ, τ 2+τ 2 2) and N(τ 2Δ/τ2 2,τ2 (τ 2+τ 2 2)/τ 2), respectively, and Z = (τ 2Y + τ2 2Y 2)/(τ 2 + τ 2 2) and Z 2 = τ2 2(Y 2 Y )/(τ 2 + τ 2 2), we have { P (τ 2 + τ 2 2 )[(γ + γ)z +{2 (γ + γ)}z 2 ] 2τ 2 Δ, } Z > Z 2,γ <n /(n + n 2 ) { = P [(γ + γ)τ 2 {2 (γ + γ)}τ 2 2 ]Y + 2τ2 2 (Y 2 τ 2 Δ/τ2 2 ), } Y >,γ <n /(n + n 2 ) = E Φ [{2 (γ + γ)}τ 2 2 (γ + γ)τ2 ]Y I Y 2τ τ 2 τ 2 + τ 2 2 > I γ<n /(n +n 2 ), where expectation is taken with respect to Y, s 2 and s2 2. Since {2 (γ + γ)}τ 2 2 (γ + γ)τ 2 >, we have the desired result. 5 Numerical results Here, we illustrate the behavior of the proposed estimators by numerical evaluation for Case-II, where both means and variances are ordered. We set γ = n s 2 2 /(n s 2 2 +n 2s 2 ) and γ = n /(n + n 2 ) when γ<n /(n + n 2 ). Then, ˆμ i (γ + ) is given by { ˆμ OS ˆμ CS =, if s2 s2 2 n min{ X, n +n 2 X + n 2 n +n 2 X 2 }, if s 2 > s2 2 and { ˆμ OS ˆμ CS 2 2 =, if s2 s2 2 n max{ X 2, n +n 2 X + n 2 n +n 2 X 2 }, if s 2 > s2 2, which are proposed by Chang and Shinozaki (22). For various values of n, n 2,Δ,σ2 2 and σ 2 =, we compare ˆμCS i with ˆμ i OS, and have numerically evaluated MPN μi ( ˆμ i CS, ˆμ i OS ), i =, 2. To compare ˆµ CS with ˆµ OS, we have numerically evaluated MPN µ ( ˆµ CS, ˆµ OS ), and also P{ X > X 2, s 2 > s2 2 }, which are all given in Table 2.

17 Estimation of two ordered normal means under modified Pitman nearness 879 MPNμ ( ˆμCS, ˆμOS ), MPN μ2 ( ˆμCS 2, ˆμOS 2 ), MPN Table 2 µ( ˆµ CS, ˆµ OS ) and P( X > X2, s 2 > s2 2 ) n n2 Δ σ 2 2 = σ 2 2 = 2 σ 2 2 = 4 a b c d a b c d a b c d (.33) (.24) (.78) (.254) (.26) (.59) (.98) (.72) (.4) (.48) (.4) (.22) (.6) (.2) (.5) (.73) (.87) (.89) (.25) (.3) (.52) (.88) (.3) (.44) (.32) (.79) (.36) (.86) (.57) (.29) (.5) (.39) (.22) (.28) (.25) (.7) (.228) (.89) (.2) (.63) (.67) (.6) (.6) (.47) (.2) (.62) (.3) (.9) (.33) (.8) (.6) (.5) (.) (.4) (.328) (.25) (.85) (.26) (.23) (.65) (.99) (.77) (.45) (.44) (.43) (.26)

18 88 Y.-T. Chang, N. Shinozaki Table 2 continued n n2 Δ σ 2 2 = σ 2 2 = 2 σ 2 2 = 4 a b c d a b c d a b c d (.99) (.2) (.8) (.65) (.86) (.9) (.272) (.32) (.49) (.95) (.2) (.4) (.27) (.75) (.33) (.74) (.52) (.26) (.39) (.34) (.9) (.8) (.2) (.4) (.25) (.79) (.3) (.64) (.57) (.) (.93) (.37) (.7) (.45) (.22) (.5) (.8) (.) (.3) (.6) (.5) (.2) MPNμ ( ˆμCS, ˆμOS ), b : MPN μ2 ( ˆμCS 2, ˆμOS 2 ), c : MPN a : µ( ˆµ CS, ˆµ OS ), d : P( X > X2, s 2 > s2 2 )

19 Estimation of two ordered normal means under modified Pitman nearness 88 From Table 2, we see that MPN μ ( ˆμ CS, ˆμOS ) > /2 when Δ = and MPN μ ( ˆμ CS, ˆμOS ) gets larger as σ 2 2 increases. However, MPN μ ( ˆμ CS, ˆμOS ) gets smaller fast as Δ increases. We see that MPN μ2 ( ˆμ CS 2, ˆμOS 2 ) gets larger as Δ increases or σ2 2 increases. We also see that MPN µ( ˆµ CS, ˆµ OS ) gets larger as σ2 2 increases, but gets smaller as Δ increases. 6 Conclusion In this paper, we have dealt with the problem of estimating two ordered normal means under modified Pitman nearness criterion when the order restriction on variances is present and not present. For the case where the order restriction on variances is not present, it is shown that the most critical case for ˆμ i (γ ) to improve upon X i is the one when μ = μ 2 and that the problem of improving upon X i reduces to the one of a common mean. This result is similar to the one given in Theorems 2. and 2.2 of Oono and Shinozaki (25) when MSE is the criterion. For the case where the order restriction on variances, σ 2 σ 2 2, is present, it is shown that we can improve upon ˆμ 2 (γ ) by ˆμ 2 (γ + ) by suitably choosing γ +. Although such an improvement is not possible for ˆμ (γ ), it is shown that ( ˆμ (γ + ), ˆμ 2 (γ + )) improves upon ( ˆμ (γ ), ˆμ 2 (γ )) for suitably chosen γ + in the simultaneous estimation of μ and μ 2. These results are similar to those given in Theorems 3., 3.3 and 4. of Chang et al. (22) when MSE or stochastic dominance is concerned. Thus, we have confirmed that similar results are obtained under modified Pitman nearness criterion to those obtained under MSE or stochastic dominance criterion. Although we have not succeeded in giving an illustrative example, one possible inconsistency among the criterions is suggested in Remark 5 which may occur due to nontransitivity of the Pitman nearness criterion. 7 Appendix A: A proof of Lemma Put k = d/(2c), then ckδ dδ = dδ/2. For any <ɛ</2, we choose sufficiently large Δ> so that Φ( dδ/2) </2 ɛ and kδ g(y)dy <ɛ kδ g(y)dy. To see that this is possible, we only need to show that lim Δ kδ g(y)dy kδ g(y)dy =, which we can verify by applying L Hospital s rule. Therefore, for sufficiently large Δ>, we have Φ(cy dδ)g(y)dy = kδ Φ(cy dδ)g(y)dy + kδ Φ(cy dδ)g(y)dy

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